Nonlinear Particle Acceleration at Nonrelativistic Shocks

1. Nonlinear Particle Acceleration at Nonrelativistic Shocks Don Ellison, North Carolina State University • Magnetic Field Amplification (MFA) from cosmic ray streaming instability • Emphasize nonlinear connection between : • First-order Fermi Particle acceleration • Shock structure • Production of magnetic turbulence • Calculation of diffusion coefficient from turbulence • Influence of amplified B-field on maximum CR energy • Particular emphasis on role of escaping particles. • Only discuss non-relativistic shocks This is NOT a formal review. Magnetic Field Amplification in shock acceleration is an active field with work being done by many people.

2. Important Points: • Collisionless shocks and the nonthermal particles they produce are widespread in astrophysics(and they are important) • In some sources, a sizable fraction of energy budget is in relativistic particles ! • Diffusive Shock Acceleration (DSA) mechanism is well-studied • Works as expected in some sources (e.g. Earth bow shock, Interplanetary shocks) • DSA is inherently efficient ! • In order for DSA to work, shocks must self-generate magnetic turbulence. • Magnetic field most important parameter in DSA • There is evidence that DSA amplifies turbulent magnetic fields by large factors: B/B >> 1(e.g., Tycho; B-field in Cas A >500 G) • High acceleration efficiency means Magnetic Field Amplification (MFA) and Diffusive Shock Acceleration (DSA) are coupled and must be treated self-consistently

3. Test-particle power law for Non-relativistic shocks (Krymskii 76; Axford, Leer & Skadron 77; Bell 78; Blandford & Ostriker 78): Power law index is: Independent of any details of diffusion  Independent of shock Obliquity (geometry) But, for Superthermal particles only  u0 is shock speed Ratio of specific heats, , along with Mach number, determines shock compression,r For high Mach number shocks:  So-called “Universal” power law from shock acceleration

4. BUT clearly Not so simple! Consider energy in accelerated particles assuming NO maximum momentum cutoff and r ~ 4(i.e., high Mach #, non-rel. shocks) Diverges if r = 4 But If produce relativistic particles < 5/3 compression ratio increases The spectrum is harder  Worse energy divergence Must have high energy cutoff in spectrum to obtain steady-state  particles must escape at cutoff !! But, if particles escape, compression ratio increases even more . . . Acceleration becomes strongly nonlinear with r >> 4 !! ►Bottom line:Strong shocks will be efficient accelerators with large comp. ratios even if injection occurs at modest levels (1 ion in 104)

5. Efficient particle acceleration  Amplification of magnetic fields

6. Evidence for High magnetic fields in SNRs (all indirect): • Broad-band fits: Same distribution of electrons produces synchrotron radio and inverse-Compton TeV -rays • 2) Spectral curvature in continuum spectra: prediction of NL shock acceleration • 3) Sharp X-ray edges: High B  large synch losses  short electron lifetime and short diffusion lengths  narrow X-ray structures. Bottom line: Inferred B-fields(200-500 G)are much larger than can be expected from simple compression of BISM Bshock >> 10 G x 4 ~ 40 G Amplification factor ~ 5 -- 50 Note: Evidence for Bshock >> compressed BISM reasonably convincing, but still room for doubt

7. Sharp X-ray synchrotron edges in SNRs : one piece of evidence for high Magnetic fields Tycho’s Supernova Remnant Sharp edge X-ray edges Blue is synchrotron emission from TeV electrons. Chandra Radial cuts: Sharp decline  high B-field Warren et al 2005 Tycho’s SNR, 4-6 keV surface brightness profiles at outer blast wave (non-thermal emission)

8. Tycho’s SNR Cassam-Chenai et al. 2007 Tycho’s SNR Radio edge not sharp  magnetic field is large: Bds~ 100 – 300 G (Note: authors more conservative in conclusions) Radio synch X-ray synch Ironically, evidence for large B-fields and MFA is obtained exclusively from radiation from electrons, butThe NL processes that produce MFA are driven by efficient acceleration of protons or other ions

9. Nonlinear coupling of DSA and MFA is a difficult plasma physics problem • Strong turbulence (or dissipation) cannot yet be treated analytically • Observations of shocks in heliosphere : • Self-generated turbulence is seen in heliospheric shocks, BUT • Weak & small heliospheric shocks don’t produce relativistic particles with high enough efficiency for MFA (as seen in SNRs) to be apparent • Particle-in-Cell (PIC) simulations of non-relativistic shocks (e.g., SNRs): • To model SNRs, require acceleration of non-relativistic particles to relativistic energies in non-relativistic shock  hard to do with PIC • PIC size and run-time requirements beyond current capabilities, but • PIC simulations are essential to understand magnetic field production and thermal particle injection • To make progress must use approximate methods: • Monte Carlo (Vladimirov, Ellison & Bykov 2006,2008) • Semi-analytic, kinetic technique based on diffusion-convection approximation (Amato & Blasi & co-workers) Here I discuss Monte Carlo work done with Andrey Vladimirov (NCSU) and Andrei Bykov (St. Petersburg)

10. Requirements for PIC simulations to do “entire” SNR problem. That is, go from injection at keV to TeV energies in non-relativistic shock Energy range: Length scale (number of cells in 1-D): Run time (number of time steps): Problem difficult because TeV protons influence injection and acceleration of keV protons and electrons: NL feedback between TeV & keV Plus, must do PIC simulations in 3-D (Jones, Jokipii & Baring 1998) PIC simulations will only be able to treat limited, but very important, parts of problem, i.e., initial B-field generation, test-particle injection To cover full dynamic range, must use approximate methods: Monte Carlo, Semi-analytic (e.g., Berezhko & co-workers; Blasi & co-workers)

11. shock flow speed, u0 u2 X Upstream DS Blast wave, i.e., Forward Shock Shocks set up converging flows of ionized plasma Some of the most energetic particles leave at “Free escape boundary” SN explosion VDS Vsk = u0 Convert to shock rest frame charged particle moving through turbulent B-field FEB Post-shock gas  Hot, compressed, dragged along with speed VDS < Vsk u2 = Vsk - VDS Particles make nearly elastic “collisions” with magnetic field  gain energy when cross shock  bulk kinetic energy of converging flows is put into individual particle energy rtot=u0/u2

12. Shock Structure and particle distributions in nonlinear DSA: plot: p4 f(p) vs. p Flow speed, u test particle shock Test-particle power law for superthermal particles only. No normalization subshock modified shock X upstream diffusion length If acceleration is efficient, shock becomes smooth from backpressure of CRs: High momentum particles “feel” a larger compression ratio  this produces a concave spectrum Injection at subshock, and maximum momentum,must be treated self-consistently Highest energy particles must escape from the shock in steady state

13. Main features of NL-DSA(Concave spectrum, Compression ratio > 7, Decrease in temperature of shocked plasma as acceleration efficiency increases) result from momentum dependence of diffusion coefficient. • If D(p) increases rapidly enough with momentum, these features occur regardless of details of wave-particle interactions. (This has been known for some time, e.g., Eichler 1984, ApJ, V. 277) • Why are details of diffusion coefficient, D(p), important? • D(p) determines injection of thermal particles  may set overall acceleration efficiency  may determine if NL effects occur at all • The production of magnetic turbulence that creates D(p) may also produce strong Magnetic Field Amplification (MFA) • If MFA occurs, the maximum particle energy a shock can produce will increase • The number and spectral shape of escaping particles will be a strong function of the detailed form of D(p) • Obliquity effects will depend on details • Electron to proton injection ratio will depend critically on diffusion details

14. Magnetic Field Amplification (MFA) in Nonlinear Diffusive Shock Acceleration using Monte Carlo methods Work done with Andrey Vladimirov & Andrei Bykov Discuss Non-relativistic shocks only here

15. How do you start with BISM 3 G and end up with B  500 G at the shock? • Basic assumptions: • Large B-fields exist and efficient shock acceleration produces them • Assume cosmic ray streaming instability is responsible, but hard to model correctly  difficult plasma physics (e.g., non-resonant interactions etc) • Connected to efficient CR production, so nonlinear effects essential • Make approximations to estimate effect as well as possible Bell & Lucek 2001  apply Q-linear theory when B/B >> 1; Bell 2004  non-resonant streaming instabilities Amato & Blasi 2006; Blasi, Amato & Caprioli 06,08; Vladimirov, Ellison & Bykov 2006, 2008 } calculations coupled to nonlinear particle accel. See references for details

16. Phenomenological approach: Growth of magnetic turbulence driven by cosmic ray pressure gradient (so-called streaming instability) e.g., McKenzie & Völk 1982 energetic particle pressure gradient as function of position, x, and momentum, p growth of magnetic turbulence energy density, W(x,k), as a function of position, x, and wavevector, k VG(x,k) parameterizes a lot of complicated plasma physics Make approximations for VG and proceed (if quasi-linear approximation applies, VG is Alfvén speed) Determine diffusion coefficient, D(x,p), from W(x,k) Use diffusion coefficient in Monte Carlo simulation Iterate

17. Once turbulence, W(x,k), is determined from CR pressure gradient, determine diffusion coefficient from W(x,k). Must make approximations here: Bohm diffusion approximation: Find effective Beff by integrating over turbulence spectrum (Vladimirov, Ellison & Bykov 06) Resonant diffusion approximation (e.g., Bell 1978; Amato & Blasi 06): Hybrid approach – Non-resonant approximation: In progress For a particle of momentum, p, have waves with scales larger and smaller than gyro-radius. How is diffusion coefficient determined?

18. Determine steady-state, shock structure with iterative, Monte Carlo technique Upstream Free escape boundary Unmodified shock with r = 4 Self-consistent, modified shock with rtot ~ 11 (rsub~ 3) Flow speed Momentum Flux conserved (within few %) Energy Flux (only conserved when escaping particles taken into account) Position relative to subshock at x = 0 [ units of convective gyroradius ]

19. Upstream Free escape boundary Energy flux in Cosmic Rays Total Energy flux Energy flux in thermal particles Energy flux in Escaping particles Beff Beff [G] Effective, amplified magnetic field, ~ 100 x upstream field Position relative to subshock at x = 0 [ units of convective gyroradius]

20. Energy flux in Cosmic Rays Total Energy flux ~ 50% of energy flux in CR spectrum ~ 35% of energy flux in escaping particles Total acceleration efficiency ~ 85% Position relative to subshock at x = 0 [ units of convective gyroradius] When the acceleration is efficient, a large fraction of energy ends up in escaping particles

21. Escaping particles in Nonlinear DSA: • Highest energy particles must scatter in self-generated turbulence. • At some distance from shock, this turbulence will be weak enough that particles freely stream away. • As these particles stream away, they generate turbulence that will scatter next generation of particles • In steady-state DSA, there is no doubt that the highest energy particles must decouple and escape – No other way to conserve energy. • In any real shock, there will be a finite length scale that will set maximum momentun, pmax. Above pmax, particles escape. • Lengths are measured in gyroradii, soB-field and MFA importantly coupled to escape and pmax • The escape reduces pressure of shocked gas and causes the overall shock compression ratio to increase (r > 7 possible). • Even if DSA is time dependent and has not reached a steady-state, the highest energy particles in the system must escape. • In a self-consistent shock, the highest energy particles won’t have turbulence to interact with until they produce it. • Time-dependent calculations (i.e., PIC sims.) needed for full solution.

22. Monte Carlo results for Bohm vs. Resonant approximations for diffusion coefficient Preliminary results: Andrey Vladimirov, Ellison & Andrei Bykov, in preparation

23. Shock structure, i.e., Flow speed vs. position Show distributions and wave spectra at various positions relative to subshock DS upstream subshock Position relative to subshock at x = 0 [ units of convective gyroradius]

24. Bohm approx. p4 f(p) k W(k,p) D(x,p)/p upstream W(k,p) Iterate: f(p) D(x,p)

25. Resonant approx. p4 f(p) k W(k,p) D(x,p)/p Seed turbulence ∝ 1/k Diffusion in resonantly amp. turb. as well as compressed seed turb. Diffusion in non-amplified but compressed seed turb. No particles resonant here D(x,p) very different from Bohm case

26. Red: Bohm Blue: Resonant k W(k,p) p4 f(p) DS DS near FEB near FEB Diffusion coefficient Flow speed D(x,p) is very different in the two cases, BUT, the shock structure & amplified B-field are adjusting to compensate for changes in D(x,p). Downstream, the particle distributions are very similar.Near the free escape boundary, large difference occur. Sold curves : downstream Dashed curves : upstream near FEB Beff

27. Energy flux in shock frame : Zero indicates isotropic flux. +1 indicates total incoming energy flux Red: Bohm Blue: Resonant Energy flux calculated downstream from the shock ~50% in CRs Escaping particles ~35% of total energy flux escapes out front of shock Importance of escaping particles discussed in recent paper: Caprioli, Blasi & Amato 2008 Our Monte Carlo results for nonlinear MFA are reasonably consistent with semi-analytic results of Blasi, Amato & co-workers

28. Shocks with and without B-field amplification Monte Carlo Particle distribution functions f(p) times p4 The maximum CR energy a given shock can produce increases with B-amp BUT Increase is not as large as downstream Bamp/B0 factor !! protons No B-amp B-amp p4 f(p) For this example, Bamp/B0 = 450G/10G = 45 but increase in pmax only ~ x5 Maximum electron energy will be determined by largest B downstream. Maximum proton energy determined by some average over precursor B-field, which is considerably smaller All parameters are the same in these cases except one has B-amplification

29. Switch gears from a Monte Carlo model of a steady-state, plane shock, to a spherically symmetric model of an expanding SNR Use semi-analytic model for nonlinear DSA from P. Blasi and co-workers Combined in VH-1 hydro code (from J. Blondin) No MFA in the following examples

30. 1-D CR-hydro model couples eff. DSA to SNR hydrodynamics Shocked ISM material : Weak X-ray lines; Strong Radio SNR Forward Shock e.g. Ellison, Decourchelle & Ballet 2004 Reverse Shock Shocked Ejecta material :Strong X-ray emission lines, but expect no radio if B is diluted progenitor field Contact Discontinuity

31. 1-D CR-hydro model couples eff. DSA to SNR hydrodynamics Shocked ISM material : Weak X-ray lines; Strong Radio SNR Forward Shock Use semi-analytic model for nonlinear DSA from P. Blasi and co-workers combined in VH-1 hydro code (J. Blondin) Reverse Shock Chandra observations of Tycho’s SNR XMM X-ray obs. of SN 1006 Kepler’s SNR Radio obs. DeLaney et al., 2002 (Warren et al. 2005) Rothenflug et al. (2004)

32. Look at acceleration efficiency in SNR over 104 yr. Energy is divided between CRs that stay in the remnant and those that escape Escaping particles don’t suffer adiabatic losses. Cosmic Rays that remain in SNR do suffer losses  escaping particles will dominate energy budget if DSA is efficient. Very efficient DSA Total Accel. Efficiency (frac.) CRs in SNR Escaping particles Total Energy / ESN Escaping particles CRs in SNR Escaping particles dominate energy budget. Most SN explosion energy ends up in escaping particles ! Work in progress

33. Less efficient DSA Escaping particles don’t suffer adiabatic losses. CRs that remain in SNR do suffer losses Here, CRs in SNR dominate energy budget Total CRs in SNR Escaping particles Total CRs in SNR Escaping particles Work in progress Note: work in progress means I’m not sure I’m right

34. Effect of escaping particles on nearby mass distributions: One-dimension SNR model in a 3-D box with arbitrary mass distribution(Lee, Kamae & Ellison 2008) Protons escaping from forward shock impact nearby molecular cloud 3-D simulation box

35. Lee et al 2008 Protons just behind the blast wave shock Escaping protons 9 pc away from center of SNR Just before impacting molecular cloud

36. Line-of-sight projections Simulation of SNR in 3-D box with arbitrary mass distribution Highest energy CRs leave the outer shock and propagate to nearby material, e.g. a dense molecular cloud Lee, Kamae & Ellison 2008 GeV TeV HESS: SNR Vela Jr.

37. Conclusions: • Magnetic field amplification (MFA) is intrinsically nonlinear  must be calculated self-consistently with shock structure • Until exact analytic descriptions of strong turbulence become available, must use approximate methods to study MFA • Monte Carlo simulations • Semi-analytic methods • In principle, can solve problem completely with PIC simulations. • However, difficult for non-relativistic shocks • Critical problems – thermal injection, initial creation of B-fields, etc., can be addressed with current PIC simulations • If shock acceleration is efficient, escaping particles will be important. • These will strongly influence wave generation and must be considered in models for CR production, TeV emission • If MFA is important in SNRs, it should be important in other systems with strong shocks (GRBs, radio jets, shocks in galaxy clusters) Don Ellison (NCSU) Talk at Cracow meeting, Oct 2008

38. Supplementary slides follow Don Ellison (NCSU) Talk at Cracow meeting, Oct 2008

39. Green line is contact discontinuity (CD) CD lies close to outer blast wave determined from 4-6 keV (non-thermal) X-rays Chandra observations of Tycho’s SNR(Warren et al. 2005) No acceleration FS FS CD RS Morphology: Strong evidence for Efficient production of cosmic ray ions at outer shock with compression ratio > 4 Efficient DSA acceleration 2-D Hydro simulationBlondin & Ellison 2001

40. Broad-band continuum emission from SNRs Berezhko & Voelk (2006) model of SNR J1713 curvature in synchrotron emission -ray HESS data fit with pion-decay from protons. Assumes large B-field X-ray radio

41. Individual particle trajectories Monte Carlo simulation (Baring, Ellison & Jones 1994) PIC simulation (Spitkovsky 2008) upstream DS upstream speed  downstream Thermal Leakage Injection Assumption: If thermal leakage is the primary injection process, this can be meaningfully described with Monte Carlo methods Note: This is only presented as a illustration. The shocks considered here (Spitkovsky simulation and Monte Carlo results) have extremely different parameters and I’m not trying to compare them directly.

42. Uncertainties in data well below the knee as well: Cosmic ray data below the knee (<1015 eV) are from balloons. These measurements are difficult !! knee 2005 Antoni et al. (KASCADE) AstroPart Phys. 2005 KASCADE is ground-based A power law can be drawn through CR data BUT, is there room for structure below the knee ?? Do individual SNRs, noticeably, contribute to all particle spectrum?? The presence of TeV electrons in CRs shows there must be a source within 100 pc

43. Cartoonof what might see from nearby sources. Will structure appear in CR spectra with more sensitive observations ? knee 2005 Antoni et al. (KASCADE) AstroPart Phys. 2005 SNR 2 SNR 1 Bottom line: Need more observations at all energies, including balloon-based below the knee

44. What does a heliospheric shock look like? shock crosses spacecraft Earth bow shock observed by AMPTE spacecraft (Ellison, Moebius & Paschmann 1990) Spacecraft give great deal of information at one point. Little or no global information time of day

45. Real shocks inject and accelerate thermal ions: AMPTE observations of diffuse ions at Q-parallel Earth bow shock H+, He2+, & CNO6+ Observed during time when solar wind magnetic field was nearly radial. Earth Bow Shock Ellison, Mobius & Paschmann 90 DS UpS DS Critical range for injection Observe injection of thermal solar wind ions at Quasi-parallel bow shock Modeling suggests nonlinear effects important

46. Observed acceleration efficiency is quite high: Dividing energy 4 keV gives 2.5% of proton density in superthermal particles, and >25% of energy flux crossing the shock put into superthermal protons Ellison, Mobius & Paschmann 90 Note: Acceleration of thermal electrons much less likely in heliospheric shocks Superthermal electrons routinely seen accelerated by heliospheric shocks, but In general, heliosphere shocks are seen NOT to accelerate thermal electrons Maxwellian

47. Real shocks, even oblique ones, inject thermal ions: ULYSSES (SWICS) observations of solar wind THERMAL ions injected and accelerated at a highly oblique Interplanetary shock Interplanetary shock Baring etal 1997 θBn=77o Bn

48. Real shocks, even oblique ones, inject thermal ions: ULYSSES (SWICS) observations of solar wind THERMAL ions injected and accelerated at a highly oblique Interplanetary shock Monte Carlo modeling implies strong scattering  ~3.7 rg Simultaneous H+ and He2+ data and modeling supports assumption that particle interactions with background magnetic field are nearly elastic Essential assumption in DSA Interplanetary shock Baring etal 1997 θBn=77o Smooth injection of thermal solar wind ions but much less efficient than Bow shock Critical range for injection

49. Self-generated turbulence at weak IPS Baring et al ApJ 1997

50. Another heliospheric shock: Protons Interplanetary Shock Obs. With GEOTAIL, 21 Feb 1994 Shimada, Terasawa, etal 1999 0.09 keV Electrons One of the very few examples where thermal electrons were observed to be injected and accelerated at heliospheric shocks Most observations of heliospheric shocks do not show the acceleration of thermal electrons 38 keV